Strong colorings over partitions

Let p:[\kappa]^2\to \theta be a partition of all unordered pairs from a cardinal \kappa to \theta pieces. A coloring f:[\kappa]^2\to \lambda is *strong over *p if for every A\subseteq \kappa with |A|=\kappa there is some i=i(A)<\theta such that ran (f\restriction ([A]^2\cap p^{-1}(\{i\}))=\lambda.

The partition symbol for asserting the existence of a strong \lambda-coloring on \kappa over a partition p is
\kappa\not\longrightarrow_p[\kappa]^2_\lambda.

In the talk we shall define more strong coloring symbols, like Pr_1 and Pr_0 and sketch the proofs of the following results:

1. Strong colorings over finite partitions exist in ZFC whenever they exist without partitions.
2. Instances of the GCH and of the SCH imply the existence of strong colorings over infinite partitions.
3. Whether for every countable partition of p:[\omega_1]^2\to \omega there is a strong \aleph_1-coloring over it, is independent over ZFC + \neg CH.

These are joint results with Bill Chen and Juris Steprans.

Recoding is now available.